This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A244650 #17 Jul 04 2014 07:19:55
%S A244650 1,9,55,290,1430,6827,32083,149665,696130,3236140,15055750,70149880,
%T A244650 327464665,1531766715,7180234915,33728718755,158765477150,
%U A244650 748819793990,3538574254840,16752185111615,79445373603241,377382842713434,1795459769465370,8554888685073420,40819261337588995
%N A244650 G.f. A(x) satisfies: Sum_{k=0..n} [x^k] A(x)^n = binomial(5*n,2*n).
%C A244650 Compare to Sum_{k=0..n} [x^k] 1/(1-x)^(4*n) = binomial(5*n,n).
%C A244650 Compare to Sum_{k=0..n} [x^k] 1/((1-x)*(1-2*x)^2)^n = binomial(4*n,2*n).
%H A244650 Paul D. Hanna, <a href="/A244650/b244650.txt">Table of n, a(n) for n = 0..120</a>
%F A244650 Recurrence: n*(n+5)*a(n) = (n+1)*(7*n+20)*a(n-1) - (n+2)*(11*n+15)*a(n-2) + 5*(n+1)*(n+2)*a(n-3). - _Vaclav Kotesovec_, Jul 04 2014
%F A244650 a(n) ~ 5^(n+11/2) / (32*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Jul 04 2014
%e A244650 G.f.: A(x) = 1 + 9*x + 55*x^2 + 290*x^3 + 1430*x^4 + 6827*x^5 +...
%e A244650 ILLUSTRATION OF INITIAL TERMS.
%e A244650 If we form an array of coefficients of x^k in A(x)^n, n>=0, like so:
%e A244650 A^0: [1], 0, 0, 0, 0, 0, 0, ...;
%e A244650 A^1: [1, 9], 55, 290, 1430, 6827, 32083, ...;
%e A244650 A^2: [1, 18, 191], 1570, 11105, 71294, 428452, ...;
%e A244650 A^3: [1, 27, 408, 4569], 42390, 345546, 2564272, ...;
%e A244650 A^4: [1, 36, 706, 10016, 115211], 1142108, 10130498, ...;
%e A244650 A^5: [1, 45, 1085, 18640, 256055, 2992934], 30938150, ...;
%e A244650 A^6: [1, 54, 1545, 31170, 497970, 6708456, 79254029],...; ...
%e A244650 then we can illustrate how the sum of the coefficients of x^k, k=0..n, in A(x)^n (shown above in brackets) equals C(5*n,2*n):
%e A244650 C( 0, 0) = 1 = 1;
%e A244650 C( 5, 2) = 1 + 9 = 10;
%e A244650 C(10, 4) = 1 + 18 + 191 = 210;
%e A244650 C(15, 6) = 1 + 27 + 408 + 4569 = 5005;
%e A244650 C(20, 8) = 1 + 36 + 706 + 10016 + 115211 = 125970;
%e A244650 C(25,10) = 1 + 45 + 1085 + 18640 + 256055 + 2992934 = 3268760;
%e A244650 C(30,12) = 1 + 54 + 1545 + 31170 + 497970 + 6708456 + 79254029 = 86493225; ...
%o A244650 (PARI) /* By Definition (slow): */
%o A244650 {a(n)=if(n==0, 1, ( binomial(5*n,2*n) - sum(k=0, n, polcoeff(sum(j=0, min(k, n-1), a(j)*x^j/1!)^n + x*O(x^k), k)))/n)}
%o A244650 for(n=0, 20, print1(a(n), ", "))
%o A244650 (PARI) /* Faster, using series reversion: */
%o A244650 {a(n)=local(B=sum(k=0, n+1, binomial(5*k,2*k)*x^k)+x^3*O(x^n), G=1+x*O(x^n));
%o A244650 for(i=1, n, G = 1 + intformal( (B-1)*G/x - B*G^2)); polcoeff(x/serreverse(x*G), n)}
%o A244650 for(n=0, 30, print1(a(n), ", "))
%Y A244650 Cf. A232606, A232683, A232687, A244650, A244651, A244652, A244653, A244654.
%K A244650 nonn
%O A244650 0,2
%A A244650 _Paul D. Hanna_, Jul 03 2014